This question is motivated from my last question here. I wonder if one has a ring A and an over-ring of this ring say B, and if we know that B is a projective A-module can we have a particular idea of how Spec B would look like if we know how Spec A would look like?

This question does make sense to me. Because for instance given a local ring A, then B's are some form of copies of A. If A were zero dimensional reduced and commutative then Spec B would look like copies of clopen subsets of Spec A (because projective modules over von Neumann regular rings are isomorphic to direct sum of principal ideals). So what else do we know?

Initially we could assume this to be so. But for the results i mentioned they need not be (modules over local rings being free or over von Neumann regular rings being direct sums of principal ideals is a general case)..except the word "copies" may not be appropriate for infinitely generated case
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Jose CapcoNov 17 '09 at 18:39

Well, if we have Spec k as a scheme, we still have k by taking global sections of the structure sheaf, so we at least know what a K-algebra looks like, but "B is a k-algebra" doesn't give us much information about spec B. Agreeing with you, but clarifying for the questioner.
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Harry GindiNov 18 '09 at 1:39